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Proof step 19 from proof mod-rec-neg (previous step)

(i0 + 1) * j + j0 = j1 * j, j0 < j, j0 < 0, i0 < j + j0, i0 < j, 0 < j, 0 < j + j0, i0 < 0, j1 * j < j + j0, j1 < 0, j1 < j, j1 < j + j0 ⊦ j = 0, j0 + j = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: numeq-01 seq: ⊦ (* i2) * i = (* i1) * i + j0 ↔ (i2 - i1) * i = j0 subst: [i2, j0, i, i1] → [j1, j0, j, i0 + 1] flags: rewrite

Additional information

Used simplifier rules: ls-same: ⊦ 0 < i → (i * j < i ↔ j < 1); from specification int-mult%sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%zero-gr-add: ⊦ 0 < j → (0 < i * j + j ↔ 0 ≤ i); from specification int-mult%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%succ-01: ⊦ i + 1 ≤ j ↔ i < j; from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs Used terms: j1, j0, j, i0 + 1